A1 — Estimating, Finding Landmark Lengths, and Rounding

Problem Strings for Fluency and Beyond · Estimating, Finding Landmark Lengths, and Rounding

Teacher notes

The first few strings in this section are designed to encourage students to estimate lengths using number relations to find landmarks (multiples of 5 and 10) in relation to the beginning and ending points of a given length. Each time you will be preparing an open number line by using a meter stick and drawing a given measured distance. For the first string below, measure a line exactly 80 centimeters long. Mark the beginning point as 0, and the endpoint as 80. There should be no other markings on the number line. Once you begin the string, show one number at a time, and ask "Where should we place this number on the number line and how do you know? After sufficient discussion time and agreement on the placement of the number, use the meter stick to check and mark the exact spot so students can see how close they came. Leave it up and move onto the next number in the string. Again ask, "Where should we place this number on the number line and how do you know?"

The purpose of the first set of strings is to encourage children to consider landmarks first and then to use them to find midpoints (multiples of fives and tens). These strings will give children an opportunity to think about the importance of 5 as a midpoint distance between the landmark decades and to use these landmarks in the base ten number system while considering the proximity of other numbers to them on an open number line. Children will begin to place numbers in relation to landmarks and/or their distance to and from landmarks. This will support their eventual understanding of the rules for rounding numbers to the nearest ten as well as to think about numbers in relationship to landmark tens and fives. This foundation serves to develop critical ideas that will support the development of rounding to the nearest 10.

Inside One Classroom: A Portion of the A1 Minilesson

Problem Strings for Fluency and Beyond · Grade 3

Julia (the teacher): Here is our first one. (Writes "40.") Ok, thumbs up when you are ready to share where you think we should place this number on the number line and how you know. (When most thumbs are up, Julia starts discussion.)

Carlos: It goes right in the middle because 40 is between 0 and 80.

Julia: Like here? (Julia marks the midpoint and labels it 40.) Wonderful! That was a fast way! Who else just knew that 40 would go right in the middle of the line? (Several hands go up.) Great! Did anyone think of it in a different way? Bella?

Bella: I tried to break the number line into groups of 10 and then counted 4 groups to know where 40 would go.

Julia: Like this? (Julia uses her hands to mark the number line into groups of ten.) Did you land at the same place as Carlos?

Bella: I'm not sure. It was hard to know exactly. But it should be because 40 is 4 tens.

Julia: Well, let's measure and see how close we came. (Julia measures with the meter stick and notes the exact place.) Wow! Our strategies were great, weren't they. Ok, let's see if we can find good estimation strategies for this one. (Writes "20.") Where should we place this number on the number line and how do you know?... Wow! Fast again! Lots of hands up. Nick?

Nick: It goes between 0 and 40 because I used Carlos' strategy. I know that 20 is half of 40.

Julia: Who else did that? (Lots of hands go up.) Ok, I was hoping that would be easy. Here's the next one then. (Julia marks the 20 and then writes 10 and all hands go up. Julia smiles.) Don't forget to show me with a thumb when you are ready. I think I need a harder one then. How about 50? (Noticing that several students are using their fingers in the air to figure out where to place the number, Julia decides to have pair talk.) Turn and talk to your elbow partner. I see lots of fingers in the air. Talk to a partner about what you are counting in the air. Is there a way to just estimate this one without counting? (After a few minutes, Julia resumes a whole group discussion.) Did anyone have an interesting partner who tried a really, fast way?

Derek: (Counting on to check what his partner has said.) Hannah said she sort of used Bella's strategy. She used the distance to 10 and added it on to 40.

Julia: Wow! So, if we use distances that we have already marked on the number line, we can think about how far away the numbers are from each other. And tens are particularly helpful, aren't they. Let's see if we can use this same strategy for the rest of the numbers in the string. What could we do to find 15?

Author's notes

Counting should not be needed for these. Julia quickly accepts a few strategies and moves on. Knowing where the challenges will be, helps her decide when to quickly move on, and when to promote discussion.

Note that Julia provides no discussion here because it is not needed. She quickly adds 10 and goes on.

Julia provides discussion here—right where it is needed.

This is a nice moment to label the strategy to promote its use. Encourage students to add on to each other's ideas and/or to give ownership of ideas to students. This values the student who thought of it and it helps others to think about using it.

The string

An open number line from 0 to 80, unmarked, used to display the start of the A1 string.